Abstract

In this dissertation, we investigate cosmological models within the framework of canonical quantum gravity based on the Wheeler–DeWitt equation with regard to whether it is possible to observe effects of quantum gravity in the Cosmic Microwave Background radiation and whether a specific class of mild singularities can be resolved by quantizing classical cosmological models in which they appear.
The first part is motivated by the fact that there are several candidates for a theory of quantum gravity and it is therefore crucial to find tests in order to figure out which theory is closest to the truth. The main problem here is that quantum-gravitational effects are highly suppressed at the energy scales one can nowadays probe in experiments. However, the inflationary phase of the universe takes place at an energy scale where effects of quantum gravity could be sizeable. During inflation one can investigate primordial cosmological perturbations that are thought to be the seed for structure formation in the early universe as well as for primordial gravitational waves. Thus they have left their imprints in the anisotropies and the polarization of the Cosmic Microwave Background radiation, which have been measured by the space observatories COBE, WMAP and Planck. We investigate to which extent quantum-gravitational effects influence these perturbations by canonically quantizing inflationary models, in which a scalar inflaton field drives the exponential expansion of the universe. At first, we analyze a simplified model, where we only add perturbations to the scalar field. Secondly, we consider scalar and tensor perturbations in a gauge-invariant way for a de Sitter universe and a generic quasi-de Sitter slow-roll model. We perform a semiclassical Born–Oppenheimer type of approximation to the Wheeler–DeWitt equation of each model and recover a Schrödinger equation for the perturbation modes as well as a modified Schrödinger equation with a quantum-gravitational correction term. From the uncorrected Schrödinger equation, we derive the usual slow-roll power spectra. The quantum-gravitational correction term leads to a modification of the power spectra on the largest scales. This effect is, however, too small to be measurable, especially in light of the statistical uncertainty due to cosmic variance, which is most prominent on large scales. We also obtain a quantum-gravitational correction to the tensor-to-scalar ratio, which is, however, much more suppressed than the second-order slow-roll corrections. Finally, we compare our results to other methods in Wheeler–DeWitt quantum cosmology and to findings in other approaches to quantum gravity.
The second part of this dissertation is based on the expectation that a quantum theory of gravity should resolve the singularities appearing in general relativity and in classical cosmology. We will focus on a specific set of cosmological singularities called type IV singularities that are of a mild nature in the sense that only higher derivatives of the Hubble parameter diverge. We model a universe with such a singularity by introducing a perfect fluid described by a generalized Chaplygin gas in the form of a scalar field, for which we consider both a standard as well as a phantom field with negative energy. After discussing the classical behavior, we can solve the Wheeler–DeWitt equation of this model analytically for a special case and can draw conclusions for the general case. We use the criterion that a singularity is avoided if the wave function vanishes in the region where the classical singularity is located. However, we obtain as a result that only particular solutions of the Wheeler–DeWitt equation of our model fulfill this criterion and therefore avoid the appearance of a type IV singularity. Lastly, we compare this result to earlier results finding an avoidance of other types of singularities and we discuss singularity resolution in other quantum gravity theories.